Noncommutative Riemannian Geometry of the Alternating Group A_4

Details Noncommutative Riemannian Geometry of the Alternating Group A_4 Noncommutative Riemannian Geometry of the Alternating Group A_4

2001-07-30

arxiv.org/abs/math/0107216v1

J.Geom.Phys. 42 (2002) 259-282

Mathematics

Quantum Algebra

28 pages Latex no figures

Scientific paper

10.1016/S0393-0440(01)00089-4

We study the noncommutative Riemannian geometry of the alternating group $A_4=(Z_2 \times Z_2)\cross Z_3$ using a recent formulation for finite groups. We find a unique `Levi-Civita' connection for the invariant metric, and find that it has Ricci-flat but nonzero Riemann curvature. We show that it is the unique Ricci-flat connection on $A_4$ with the standard framing (we solve the vacuum Einstein's equation). We also propose a natural Dirac operator for the associated spin connection and solve the Dirac equation. Some of our results hold for any finite group equipped with a cyclic conjugacy class of 4 elements. In this case the exterior algebra $\Omega(A_4)$ has dimensions $1:4:8:11:12:12:11:8:4:1$ with top-form 9-dimensional. We also find the noncommutative cohomology $H^1(A_4)=C$.

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